{"paper":{"title":"Characterization of $n$-rectifiability in terms of Jones' square function: Part I","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.CA","authors_text":"Xavier Tolsa","submitted_at":"2015-01-07T17:38:05Z","abstract_excerpt":"In this paper it is shown that if $\\mu$ is a finite Radon measure in $\\mathbb R^d$ which is $n$-rectifiable and $1\\leq p\\leq 2$, then $$\\int_0^\\infty \\beta_{\\mu,p}^n(x,r)^2\\,\\frac{dr}r<\\infty \\quad {for $\\mu$-a.e. $x\\in\\mathbb R^d$,}$$ where $$\\beta_{\\mu,p}^n(x,r) = \\inf_L (\\frac1{r^n} \\int_{\\bar B(x,r)} (\\frac{\\mathrm dist(y,L)}{r})^p\\,d\\mu(y))^{1/p},$$ with the infimum taken over all the $n$-planes $L\\subset \\mathbb R^d$. The $\\beta_{\\mu,p}^n$ coefficients are the same as the ones considered by David and Semmes in the setting of the so called uniform $n$-rectifiability. An analogous necessar"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.01569","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}